Concerning the Onsager conjecture with the critical spatial regularity, namely Hölder exponent \(\theta =\frac{1}{3}\) , there are also some interesting results. By time localized estimates and careful choice of the parameters in [1]}, T. Buckmaster constructed Hölder continuous periodic solutions with exponent \(\theta <\frac{1}{5}\)  in time-space, which for almost every time belongs to \(C_x^{\theta }\) , for any \(\theta <\frac{1}{3}\)  and is compactly temporal supported. Later, by smoothing Reynolds stress for different time intervals using different approach carefully and introducing some novel ideas in [2]} , Camillo De Lellis, László Székelyhidi and T. Buckmaster constructed Hölder continuous periodic solution which belongs to \(L^1_tC_x^\theta \) , for any \(\theta <\frac{1}{3}\)  and has compact support in time.
